T-Totals. I am going to investigate T-totals in relation to the T-number on different sized grids. I am then going to investigate the relationship between the T-total of a T-shape in 1 area of a grid to when it is translated, using any vector, to another
For this Piece of coursework I am going to investigate T-totals in relation to the T-number on different sized grids. I am then going to investigate the relationship between the T-total of a T-shape in 1 area of a grid to when it is translated, using any vector, to another area of the grid.
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A T-shape consists of five numbers, when added together theses numbers create the T-total. The T-number is the number at the bottom of the T-shape.
E.G.
- The T-total for this shape would be
7+8+9+13+18 = 55
- The T-number for this shape would be
18
Throughout this Coursework I will refer to the T-total as 'T' and the T-number as 'N'.
I started by investigating the relationships on a 5x5 grid, making sure I worked in a systematic way in order to make it easier to compare the results and discover a comparison.
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T-number (n)
T-Total (T)
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25
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30
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35
I can see from the table that as N increases by 1, T increases by 5, using this information I can begin to create a formula.
Proof
I created the T-shape, for this grid, below in order to prove my formula.
N-11
N-10
N-9
N-5
N
So, T = n-11+n-10+n-9+n-5+n
T = 5n - 35
Therefore, according to this T-shape my formula is correct.
I then worked out the formula for a 6x6 grid using the same process.
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T-number (n)
T-Total (T)
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...
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I then worked out the formula for a 6x6 grid using the same process.
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T-number (n)
T-Total (T)
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28
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38
I can see from the table that, like on the 5x5 grid, as N increases by 1, T increases by 5. So I know that the beginning of the formula is the same as the grid before.
4(n) x 5 = 70
70 - 28(T) = 42
So, T = 5n - 42
Proof
I created the T-shape, for this grid, below in order to prove my formula.
N-13
N-12
N-11
N-6
N
So, T = n-13+n-12+n-11+n-6+n
T = 5n - 42
Therefore, according to this T-shape my formula is correct.
I then worked out the formula for a 7x7 grid using the same process.
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T-number (n)
T-Total (T)
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41
I can see from the table that, like on the other grid, as N increases by 1, T increases by 5. So I know that the beginning of the formula is the same.
6(n) x 5 = 80
80 - 21(T) = 49
So, T = 5n - 49
Proof
I created the T-shape, for this grid, below in order to prove my formula.
N-15
N-14
N-13
N-7
N
So, T = n-15+n-14+n-13+n-7+n
T = 5n - 49
Therefore, according to this T-shape my formula is correct.
I then worked out the formula for a 8x8 grid using the same process.
T-number (n)
T-Total (T)
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I can see from the table that, like on the other grid, as N increases by 1, T increases by 5. So I know that the beginning of the formula is the same.
8(n) x 5 = 90
90 - 34(T) = 56
So, T = 5n - 56
Proof
I created the T-shape, for this grid, below in order to prove my formula.
N-17
N-16
N-15
N-8
N
So, T = n-13+n-12+n-11+n-6+n
T = 5n - 42
Therefore, according to this T-shape my formula is correct.
I then worked out the formula for a 9x9 grid using the same process.
T-number (n)
T-Total (T)
20
37
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42
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47
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I can see from the table that, like on the other grid, as N increases by 1, T increases by 5. So I know that the beginning of the formula is the same.
20(n) x 5 = 100
00 - 37(T) = 63
So, T = 5n - 63
Proof
I created the T-shape, for this grid, below in order to prove my formula.
N-19
N-18
N-17
N-9
N
So, T = n-19+n-18+n-17+n-9+n
T = 5n - 63
Grid Size
Formula
5 x 5
T = 5n - 35
6 x 6
T = 5n - 42
7 x 7
T = 5n - 49
8 x 8
T = 5n - 56
9 x 9
T = 5n - 63
Looking at the different formulae I can see that there is an increase by 7, and if I times the grid size by this I am left with the number that is subtracted from 5n.
For example
5 x 7 = 35
5 x 5 grid T = 5n - 35
Using this information I can create a formula that works for any grid size. It can be used to find T when given the grid size and T-number.
T = 5n - 7G
(G = grid size)
Proof
I created the T-shape, for any grid, below in order to prove my formula.
N-2G+1
N-2G
N-2G-1
N-G
N
So, T = n-2g+1+n-2g+n-2g-1+n-g+n
T = 5n - 7G
Now I am going to investigate the relationship between the T-total of a T-shape in 1 area of a grid to when it is translated, using any vector, to another area of the grid.
I will use a 10x10 grid and experiment with different vectors to see if there is a pattern.
H amount moved horizontally
V amount moved vertically
N-2G+1
N-2G
N-2G-1
N-G
N
N-2G+1 +h
N-2G+h
N-2G-1+h
N-G+h
N+h
Original T-shape T-shape moved right H
So,T-total = n-2g+1+h+n-2g+h+n-2g-1+h+n-g+h+n+h
= 5N - 7G + 5H
N-2G+1
N-2G
N-2G-1
N-G
N
N-2G+1 -Gv
N-2G-Gv
N-2G-1-Gv
N-G-Gv
N-Gv
Original T-shape T-shape moved up V
So,T-total = n-2g+1-gv+n-2g-gv+n-2g-1-gv+n-n-g-gv+n-gv
= 5N - 7G + 5GV
These can be proven by checking it within the grid below.
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N-2G+1 +1
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N-2G-10
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N-G-10
N-10
T-shape moved right 1 T-shape moved up 1
In conclusion I have found that the T-total of a vector H is
V
T = 5N - 7G + 5H - GV
